The Expected Number of Roots over The Field of p-adic Numbers

We study the roots of a random polynomial over the field of $p$-adic numbers. For a random monic polynomial with i.i.d. coefficients in ${\mathbb{Z}}_p$, we obtain an estimate for the expected number of roots of this polynomial. In particular, if the coefficients take the values $\pm 1$ with equal probability, the expected number of $p$-adic roots converges to $(p-1)\mathrm{/}(p+1)$ as the degree of the polynomial tends to $\mathrm{\infty }$.